Add the following rational expressions. $\dfrac{3x^2}{x+11}+\dfrac{-1}{2x^3}=$
We can add two rational expressions whose denominators are equal by adding the numerators and keeping the denominator the same. [Does this fit with how we add rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({x+11})\cdot({2x^3})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{3x^2}{{x+11}}+\dfrac{-1}{{2x^3}} \\\\ &=\dfrac{3x^2\cdot({2x^3})}{({x+11})\cdot({2x^3})}+\dfrac{-1\cdot({x+11})}{({2x^3})\cdot({x+11})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's add! $\begin{aligned} &\phantom{=}\dfrac{3x^2\cdot(2x^3)}{(x+11)\cdot(2x^3)}+\dfrac{-1\cdot(x+11)}{(2x^3)\cdot(x+11)} \\\\ &=\dfrac{3x^2\cdot(2x^3)-1\cdot(x+11)}{(x+11)(2x^3)} \\\\ &=\dfrac{6x^5-x-11}{(x+11)(2x^3)} \end{aligned}$ In conclusion, $\dfrac{3x^2}{x+11}+\dfrac{-1}{2x^3}=\dfrac{6x^5-x-11}{(x+11)(2x^3)}$